Numerical treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers

Authors:
Alfredo Bermúdez, Rodolfo Rodríguez and Pilar Salgado

Journal:
Math. Comp. **74** (2005), 123-151

MSC (2000):
Primary 78M10, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-04-01680-1

Published electronically:
July 20, 2004

MathSciNet review:
2085405

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Abstract: This paper deals with the finite element solution of the eddy current problem in a bounded conducting domain, crossed by an electric current and subject to boundary conditions appropriate from a physical point of view. Two different cases are considered depending on the boundary data: input current density flux or input current intensities. The analysis of the former is an intermediate step for the latter, which is more realistic in actual applications. Weak formulations in terms of the magnetic field are studied, the boundary conditions being imposed by means of appropriate Lagrange multipliers. The resulting mixed formulations are analyzed depending on the regularity of the boundary data. Finite element methods are introduced in each case and error estimates are proved. Finally, some numerical results to assess the effectiveness of the methods are reported.

**1.**A. Alonso P. Fernandes, and A. Valli,*The time-harmonic eddy current problem in general domains: solvability via scalar potentials*, Lect. Notes Comp. Sci. Engrg.,**28**(2003) 143-163.**2.**A. Alonso and A. Valli,*A domain decomposition approach for heterogeneous time-harmonic Maxwell equations*, Comput. Methods Appl. Mech. Engrg.,**143**(1997) 97-112. MR**98b:78020****3.**-,*An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations*, Math. Comp.,**68**(1999) 607-631. MR**99i:78002****4.**C. Amrouche, C. Bernardi, M. Dauge, and V. Girault,*Vector potentials in three-dimensional non-smooth domains*, Math. Meth. Appl. Sci.,**21**(1998) 823-864. MR**99e:35037****5.**A. Bermúdez, J. Bullón, and F. Pena,*A finite element method for the thermoelectrical modelling of electrodes*, Comm. Numer. Methods Engrg.,**14**(1998) 581-593.**6.**A. Bermúdez, J. Bullón, F. Pena, and P. Salgado,*A numerical method for transient simulation of metallurgical compound electrodes*, Finite Elem. Anal. Des.,**39**(4) (2003) 283-299.**7.**A. Bermúdez, R. Rodríguez, and P. Salgado,*A finite element method with Lagrange multipliers for low-frequency harmonic Maxwell equations*, SIAM J. Numer. Anal.,**40**(5) (2002) 1823-1849. MR**2004b:78017****8.**-,*Modeling and numerical treatment of boundary data in an eddy current problem*, C. R. Acad. Sci. Paris, Serie I,**335**(7) (2002) 633-638.**9.**A. Bossavit,*The computation of eddy-currents in dimension 3 by using mixed finite elements and boundary elements in association*, Math. Comput. Modelling,**15**(1991) 33-42. MR**92c:78001****10.**-,*Computational Electromagnetism. Variational Formulations, Complementarity, Edge Elements*. Academic Press. San Diego, CA, 1998. MR**99m:78001****11.**-,*Most general ``non-local'' boundary conditions for the Maxwell equation in a bounded region*, COMPEL,**19**(2000) 239-245. MR**2001f:78009****12.**A. Bossavit and J.C. Vérité,*A mixed FEM-BIEM method to solve 3-D eddy current problems*, IEEE Trans. Mag.,**18**(1982) 431-435. MR**2001f:78009****13.**M. Dauge,*Elliptic boundary value problems on corner domains: smoothness and asymptotics of solutions*, Lecture Notes in Mathematics**1341**, Springer, Berlin, 1988. MR**91a:35078****14.**V. Girault and P.A. Raviart,*Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms*. Springer-Verlag, Berlin, 1986. MR**88b:65129****15.**S.I. Hariharan and R.C. MacCamy,*An integral equation procedure for eddy current problems*, J. Comput. Phys.,**45**(1982) 80-99. MR**83g:78011****16.**R. Innvær and L. Olsen,*Practical use of mathematical models for Soderberg electrodes.*Elkem Carbon Technical Paper presented at the A.I.M.E. Conference. (1980).**17.**R.C. MacCamy and E. Stephan,*A skin effect approximation for eddy current problems*, Arch. Rational Mech. Anal.,**90**(1985) 87-98. MR**86j:78005****18.**R.C. MacCamy and M. Suri,*A time-dependent interface problem for two-dimensional eddy currents*, Quart. Appl. Math.,**44**(1987) 675-690. MR**87m:78007****19.**J.C. Nédélec,*Mixed finite elements in*, Numer. Math.,**35**(1980) 315-341. MR**81k:65125**

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Additional Information

**Alfredo Bermúdez**

Affiliation:
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain

Email:
mabermud@usc.es

**Rodolfo Rodríguez**

Affiliation:
GI$^{2}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Email:
rodolfo@ing-mat.udec.cl

**Pilar Salgado**

Affiliation:
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain

Email:
mpilar@usc.es

DOI:
https://doi.org/10.1090/S0025-5718-04-01680-1

Keywords:
Low-frequency harmonic Maxwell equations,
eddy current problems,
finite element computational electromagnetism,
Lagrange multipliers

Received by editor(s):
October 15, 2002

Received by editor(s) in revised form:
July 16, 2003

Published electronically:
July 20, 2004

Additional Notes:
This work was partially supported by Programa de Cooperación Científica con Iberoamérica, Ministerio de Educación y Ciencia, Spain.

The first author was partially supported by Xunta de Galicia grant PGIDT00PXI20701PR (Spain).

The second author was partially supported by FONDAP in Applied Mathematics (Chile).

The third author was partially supported by FEDER-CICYT grant 1FD97.0280 (Spain).

Article copyright:
© Copyright 2004
American Mathematical Society